Page 487 - Handbook Of Integral Equations

9.3-6. Systems of Volterra Integral Equations
The Laplace transform can be applied to solve systems of Volterra integral equations of the form
n x

y m (x) – K mk (x – t)y k (t) dt = f m (x), m =1, ... , n. (31)
0
k=1
Let us apply the Laplace transform to system (31). We obtain the relations
n

˜
˜
˜ y m (p) – K mk (p) ˜y k (p)= f m (p), m =1, ... , n. (32)
k=1
On solving this system of linear algebraic equations, we find ˜y m (p), and the solution of the system
under consideration becomes
1 c+i∞ px
y m (x)= ˜ y m (p)e dp. (33)
2πi
c–i∞
The Laplace transform can be applied to construct a solution of systems of Volterra equations
of the first kind and of integro-differential equations as well.
•
References for Section 9.3: V. A. Ditkin and A. P. Prudnikov (1965), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko
(1971), V. I. Smirnov (1974), K. B. Oldham and J. Spanier (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov
and Yu. I. Cherskii (1978), Yu. I. Babenko (1986), R. Gorenflo and S. Vessella (1991), S. G. Samko, A. A. Kilbas, and
O. I. Marichev (1993).

9.4. Operator Methods for Solving Linear Integral
Equations

9.4-1. Application of a Solution of a “Truncated” Equation of the First Kind
Consider the linear equation of the second kind
y(x)+ L [y]= f(x), (1)
where L is a linear (integral) operator.
Assume that the solution of the auxiliary “truncated” equation of the first kind
L [u]= g(x), (2)
can be represented in the form

u(x)= M L[g] , (3)
where M is a known linear operator. Formula (3) means that
L –1 = ML.
–1
Let us apply the operator L to Eq. (1). The resulting relation has the form

M L[y] + y(x)= M L[f] , (4)
On eliminating y(x) from (1) and (4) we obtain the equation
M [w] – w(x)= F(x), (5)
in which the following notation is used:

w = L [y], F(x)= M L[f] – f(x).
In some cases, Eq. (5) is simpler than the original equation (1). For example, this is the case if
the operator M is a constant (see Subsection 11.7-2) or a differential operator:
d
n
n–1
M = a n D + a n–1 D + ··· + a 1 D + a 0 , D ≡ .
dx
In the latter case, Eq. (5) is an ordinary linear differential equation for the function w.
If a solution w = w(x) of Eq. (5) is obtained, then a solution of Eq. (1) is given by the formula

y(x)= M L[w] .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 469